The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A372223

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A372223

Numbers in a hexagonal tiling (seen as concentric rings) which have exactly three neighbors whose difference from it is prime.
(history; published version)

#31 by Alois P. Heinz at Mon Jul 01 14:36:18 EDT 2024

STATUS

editing

approved

#30 by Alois P. Heinz at Mon Jul 01 14:36:13 EDT 2024

LINKS

Project Euler, <a href="https://projecteuler.net/problem=128">problem Problem 128 - Hexagonal Tile Differences</a>.

STATUS

proposed

editing

#29 by Antoine Mathys at Mon Jul 01 14:32:08 EDT 2024

STATUS

editing

proposed

#28 by Antoine Mathys at Mon Jul 01 14:31:58 EDT 2024

LINKS

Project Euler , <a href="https://projecteuler.net/problem=128">problem 128 - Hexagonal Tile Differences</a>, on which this sequence is based.

STATUS

approved

editing

#27 by OEIS Server at Fri Jun 28 23:14:22 EDT 2024

LINKS

Antoine Mathys, <a href="/A372223/b372223_1.txt">Table of n, a(n) for n = 1..10000</a>

#26 by N. J. A. Sloane at Fri Jun 28 23:14:22 EDT 2024

STATUS

proposed

approved

Discussion

Fri Jun 28

23:14

OEIS Server: Installed first b-file as b372223.txt.

#25 by Antoine Mathys at Fri Jun 28 10:15:58 EDT 2024

STATUS

editing

proposed

Discussion

Fri Jun 28

10:17

Antoine Mathys: Fixed it.

#24 by Antoine Mathys at Fri Jun 28 10:14:52 EDT 2024

EXAMPLE

26--25--24--23

/ \

27 12--11--10 22

/ / \ \

28 13 4---3 9 21

/ / / \ \ \

29 14 5 1 2 8 20

\ \ \ / / \

\ \ \ / / /

30 15 6---7 19 37 \ \ / /

\ \ / /

31 16--17--18 36

\ /

32---33--34---35.

.

STATUS

proposed

editing

#23 by Antoine Mathys at Fri Jun 28 05:55:30 EDT 2024

STATUS

editing

proposed

Discussion

Fri Jun 28

07:36

Michel Marcus: something went wrong in the example picture

#22 by Antoine Mathys at Thu Jun 27 22:00:27 EDT 2024

NAME

Numbers in a hexagonal tiling (seen as concentric rings) which have exactly three neighbors whose difference from it is prime.

COMMENTS

A hexagonal tile with number 1 is surrounded by a ring of six hexagonal tiles, starting at "noon" on the right and numbering the tiles 2 to 7 in a counterclockwise direction. New rings are added in the same fashion, with the next rings being numbered 8 to 19, 20 to 37, and so on (see the illustration below). By finding the difference between tile n and each of its six neighbors we shall define PD(n) to be the number of those differences which are prime. For example, working counter-clockwise around tile 8 the differences are 12, 29, 13, 1, 6, 11, 6, 1, and 1329. So PD(8)=3. In the same way, the differences around tile 17 are 1, 17, 16, 1, 10, 11, 1, 16, and 10, 17, hence PD(17) = 2. It can be shown that the maximum value of PD(n) is 3. This sequence lists all tiles for which PD(n)=3 in ascending order.

EXAMPLE

. __

__/38\__

__/39\__/61\__

__/40\__/20\__/60\__

__/41\__/21\__/37\__/59\__

/42\__/22\__/ 8\__/36\__/58\

\__/23\__/ 9\__/19\__/35\__/

/43\__/10\__/ 2\__/18\__/57\

\__/24\__/ 3\__/ 7\__/34\__/

/44\__/11\__/ 1\__/17\__/56\

\__/25\__/ 4\__/ 6\__/33\__/

/45\__/12\__/ 5\__/16\__/55\

\__/26\__/13\__/15\__/32\__/

/46\__/27\__/14\__/31\__/54\

\__/47\__/28\__/30\__/53\__/

\__/48\__/29\__/52\__/

\__/49\__/51\__/

\__/50\__/

\__/

26--25--24--23

/ \

27 12--11--10 22

/ / \ \

28 13 4---3 9 21

/ / / \ \ \

29 14 5 1 2 8 20

\ \ \ / / \

30 15 6---7 19 37 \ \ / /

31 16--17--18 36

\ /

32---33--34---35.

Using tile numbering as in A056105 (and rotated 90 degrees counterclockwise):

__

/66\__

.\__/65\__

. __/41\__/64\__

. __/42\__/40\__/63\__

__/43\__/22\__/39\__/62\

__/44\__/23\__/21\__/38\__/

/45\__/24\__/ 9\__/20\__/61\

\__/25\__/10\__/ 8\__/37\__/

/46\__/11\__/ 2\__/19\__/60\

\__/26\__/ 3\__/ 7\__/36\__/

/47\__/12\__/ 1\__/18\__/59\

\__/27\__/ 4\__/ 6\__/35\__/

/48\__/13\__/ 5\__/17\__/58\

\__/28\__/14\__/16\__/34\__/

/49\__/29\__/15\__/33\__/57\

\__/50\__/30\__/32\__/56\__/

\__/51\__/31\__/55\__/

\__/52\__/54\__/

\__/53\__/

\__/

PROG

(C++)

(C++)#include <iostream>

Discussion

Thu Jun 27

22:06

Antoine Mathys: A hexagonal tiling can be seen as a hexagonal spiral (as in A056105), or as a set of concentric hexagons, as is the case here. These have completely different structures.

In the latter case, it makes sense for the first cell of each ring to be on the same spoke. I changed the definition (and illustration) of the numbering to rotate it by 90° clockwise, so that the first cell of each ring is on the right spoke (the one of angle 0°).

If you define PD using the numbering matching a spiral, you get the following values: https://pastebin.com/4xFNLU5t
As you can see, only PD(1)=3, otherwise PD is at most 2 and the values occur in blocks, thus the positions in which PD=2 are numerous and not interesting.

The proposed sequence is an interesting property of concentric hexagons and only makes sense in this context. I checked the OEIS and this is apparently not covered anywhere.

I wish people didn't rush to judgment or made edit decisions on things they didn't take the time to understand.